An integer sequence with length n, denoted by 1,2,⋯,a1,a2,⋯,an, is generated randomly, and the probability of being 1,2,⋯,1,2,⋯,n are all 11n for each ai (=1,2,⋯,)(i=1,2,⋯,n).
Your task is to calculate the expected number of permutations 1,2,⋯,p1,p2,⋯,pn from 11 to nsuch that ≤pi≤ai holds for each =1,2,⋯,i=1,2,⋯,n.
Input
The only line contains an integer n (1≤≤50)(1≤n≤50).
Output
Output the expected number of permutations satisfying the condition. Your answer is acceptable if its absolute or relative error does not exceed 10−910−9.
Formally speaking, suppose that your output is x and the jury's answer is y. Your output is accepted if and only if |−|max(1,||)≤10−9|x−y|max(1,|y|)≤10−9.
Examples
input
Copy
2
output
Copy
1.000000000000
input
Copy
3
output
Copy
1.333333333333
input
Copy
50
output
Copy
104147662762941310907813025277584020848013430.758061352192
首先易知可能的a序列总共有\(n^n\)个。p序列一共有\(n!\)种可能（因为是1～n排列），对于每种排列，满足条件的a序列有\(n!\)个，因为对于\(p_1,p_2...p_n\)，a序列只能这样选：\([p1, n], [p2, n]...[pn, n]\)，且p1到pn是一个1到n的排列。注意因为是求期望，所以不用考虑重复的情况。
import math
n = eval(input())
print((math.factorial(n)) ** 2 / n ** n)